Angle and parallel-line proofs: fill in the missing reason

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Given: UV ∥ WY. Prove: ∠USQ and ∠SQW are supplementary. Ladder: 1. UV ∥ WY -- Given; 2. ∠USQ ≅ ∠SQY -- Alternate interior angles theorem; 3. m∠USQ = m∠SQY -- Definition of congruent angles; 4. ∠SQW and ∠SQY are supplementary -- Linear pair postulate; 5. m∠SQW + m∠SQY = 180 -- Definition of supplementary angles; 6. m∠SQW + m∠USQ = 180 -- ?; 7. ∠USQ and ∠SQW are supplementary -- Definition of supplementary angles.

2. 1.

   Ari proves the same-side interior angles theorem on train rails UV and WY. Which reason fills the blank?

   • Substitution property
   • Definition of supplementary angles
   • Transitive property
   • Linear pair postulate

3. Use this

   Given: UV ∥ WY. Prove: ∠VSQ and ∠SQY are supplementary. Ladder: 1. UV ∥ WY -- Given; 2. ∠VSQ ≅ ∠SQW -- Alternate interior angles theorem; 3. m∠VSQ = m∠SQW -- Definition of congruent angles; 4. ∠SQY and ∠SQW are supplementary -- Linear pair postulate; 5. m∠SQY + m∠SQW = 180 -- Definition of supplementary angles; 6. m∠SQY + m∠VSQ = 180 -- ?; 7. ∠VSQ and ∠SQY are supplementary -- Definition of supplementary angles.

4. 2.

   Bo proves the same-side interior angles theorem on rug stripes UV and WY. Which reason fills the blank?

   • Substitution property
   • Addition property of equality
   • Given
   • Vertical angles theorem

5. Use this

   Given: ∠USQ and ∠SQW are supplementary. Prove: UV ∥ WY. Ladder: 1. ∠USQ and ∠SQW are supplementary -- Given; 2. ∠SQW and ∠SQY are supplementary -- Linear pair postulate; 3. ∠USQ ≅ ∠SQY -- ?; 4. UV ∥ WY -- Converse of the alternate interior angles theorem.

6. 3.

   Kira checks that tent poles UV and WY are parallel, using a cord HD. Which reason fills the blank?

   • Alternate interior angles theorem
   • Congruent complements theorem
   • Vertical angles theorem
   • Congruent supplements theorem
7. 4.

   Luz checks that a door's edges UV and WY are parallel, using a ribbon HD. Which reason fills the blank?

   • Congruent complements theorem
   • Vertical angles theorem
   • Congruent supplements theorem
   • Alternate interior angles theorem

8. Use this

   Given: UV ∥ WY. Prove: ∠USQ and ∠SQW are supplementary. Ladder: 1. UV ∥ WY -- Given; 2. ∠USQ ≅ ∠WQD -- Corresponding angles postulate; 3. m∠USQ = m∠WQD -- Definition of congruent angles; 4. ? -- Linear pair postulate; 5. m∠SQW + m∠WQD = 180 -- Definition of supplementary angles; 6. m∠SQW + m∠USQ = 180 -- Substitution property; 7. ∠USQ and ∠SQW are supplementary -- Definition of supplementary angles.

9. 5.

   Esme proves the same-side interior angles theorem a second way, on map streets UV and WY. Which statement fills the blank?

   • ∠SQW and ∠WQD are complementary
   • ∠SQW and ∠WQD are supplementary
   • ∠USQ and ∠SQW are supplementary
   • ∠SQW ≅ ∠WQD

10. Use this

    Sketch, not to scale: line UV above line WY (U and W on the left), crossed by line HD at S on UV and at Q on WY (H at the top, D at the bottom). Marked: parallel arrows on UV and WY. Given: UV ∥ WY. Prove: ∠VSQ and ∠SQY are supplementary. Ladder: 1. UV ∥ WY -- Given; 2. ∠VSQ ≅ ∠YQD -- Corresponding angles postulate; 3. m∠VSQ = m∠YQD -- Definition of congruent angles; 4. ? -- Linear pair postulate; 5. m∠SQY + m∠YQD = 180 -- Definition of supplementary angles; 6. m∠SQY + m∠VSQ = 180 -- Substitution property; 7. ∠VSQ and ∠SQY are supplementary -- Definition of supplementary angles.

11. 6.

    Jun proves the same-side interior angles theorem a second way, on window bars UV and WY. Which statement fills the blank?

    • ∠SQY and ∠YQD are supplementary
    • ∠SQY and ∠YQD are complementary
    • ∠SQY ≅ ∠YQD
    • ∠VSQ and ∠SQY are supplementary

12. Use this

    Sketch, not to scale: rays QC, QD, QF and QG come out of point Q in that order, turning one way. Marked: right-angle boxes at ∠CQF and ∠DQG. Given: ∠CQF and ∠DQG are right angles. Prove: ∠CQD ≅ ∠FQG. Ladder: 1. ∠CQF and ∠DQG are right angles -- Given; 2. m∠CQF = 90 and m∠DQG = 90 -- Definition of right angle; 3. ? -- Angle addition postulate; 4. m∠DQF + m∠FQG = m∠DQG -- Angle addition postulate; 5. m∠CQD + m∠DQF = 90 and m∠DQF + m∠FQG = 90 -- Substitution property; 6. ∠CQD and ∠DQF are complementary, and so are ∠DQF and ∠FQG -- Definition of complementary angles; 7. ∠CQD ≅ ∠FQG -- Congruent complements theorem.

13. 7.

    Milo folds a paper fan whose folds are rays QC, QD, QF and QG. Which statement fills the blank?

    • m∠CQD + m∠DQF = 180
    • ∠CQD ≅ ∠FQG
    • m∠CQD + m∠FQG = m∠CQG
    • m∠CQD + m∠DQF = m∠CQF

14. Use this

    Given: ∠UQW and ∠VQY are right angles. Prove: ∠UQV ≅ ∠WQY. Ladder: 1. ∠UQW and ∠VQY are right angles -- Given; 2. m∠UQW = 90 and m∠VQY = 90 -- Definition of right angle; 3. m∠UQV + m∠VQW = m∠UQW -- Angle addition postulate; 4. ? -- Angle addition postulate; 5. m∠UQV + m∠VQW = 90 and m∠VQW + m∠WQY = 90 -- Substitution property; 6. ∠UQV and ∠VQW are complementary, and so are ∠VQW and ∠WQY -- Definition of complementary angles; 7. ∠UQV ≅ ∠WQY -- Congruent complements theorem.

15. 8.

    Noah draws four sun rays QU, QV, QW and QY from one center Q. Which statement fills the blank?

    • m∠VQW + m∠WQY = 180
    • m∠VQW + m∠WQY = m∠VQY
    • ∠UQV ≅ ∠WQY
    • m∠UQV + m∠WQY = m∠UQY
16. 9.

    Tala proves the converse of the alternate exterior angles theorem on quilt stripes UV and WY. Which statement and reason fill the blanks?

    • ∠VSQ ≅ ∠YQD — Transitive property
    • ∠VSQ ≅ ∠YQD — Corresponding angles postulate
    • ∠HSU ≅ ∠VSQ — Reflexive property
    • ∠VSQ ≅ ∠YQD — Symmetric property

17. Use this

    Given: ∠UQV and ∠VQW are complementary, and so are ∠WQY and ∠VQW. Prove: ∠UQV ≅ ∠WQY. Ladder: 1. ∠UQV and ∠VQW are complementary, and so are ∠WQY and ∠VQW -- Given; 2. m∠UQV + m∠VQW = 90 -- Definition of complementary angles; 3. m∠WQY + m∠VQW = 90 -- Definition of complementary angles; 4. m∠UQV + m∠VQW = m∠WQY + m∠VQW -- Substitution property; 5. ? -- ?; 6. ∠UQV ≅ ∠WQY -- Definition of congruent angles.

18. 10.

    Ren proves the congruent complements theorem on sun rays QU, QV, QW and QY. Which statement and reason fill the blanks?

    • ∠UQV ≅ ∠WQY — Congruent complements theorem
    • m∠UQV = m∠WQY — Subtraction property of equality
    • m∠UQV = m∠VQW — Subtraction property of equality
    • m∠UQV = m∠WQY — Addition property of equality